Einstein and Group Contractions

When Einstein introduced Lorentz covariance for Newtonian particle
dynamics, the question was how he could explain the fact that
Newton's law is perfectly consistent with the Galilei system. The
explanation was and still is that Lorentz transformations become
Galilei transformations in the limit of small velocity or large
velocity of light.

Erdal Inonu giving the talk at the workshop (1997).

This is of course very easy to say and write down the formulas, but
the precise group theoretical treatment was not achieved until 1953
when Erdal Inonu and Eugene Wigner published a paper on group
contractions [Proc. Nat. Aca. Sci. 39, 510 (1953)].
They explained precisely how the six generators of
the Lorentz group become the six generators of the Galilei group.
You may be interested in
Inonu's review paper given at the
Workshop on Quantum Groups, Deformations and Contractions
(Istanbul, Turkey, 1997).

In the same paper, Inonu and Wigner give a detailed exposition
of the contraction of the O(3) rotation group into the E(2) group,
namely the Euclidean group in two-dimensional space consisting
of rotations around the origin and translations in two orthogonal
directions. We shall see what effect this has in studying internal
space-time symmetries of relativistic particles.

In his 1939 paper on the inhomogeneous Lorentz group
[Ann. Math. 40, 149], Wigner showed
that the internal space-time symmetries of massive and massless particles
are isomorphic to O(3) (three-dimensional rotation group) and
E(2) (two-dimensional Euclidean group) respectively. They are known as
Wigner's little groups.

The O(3)-like symmetry for a massive particle corresponds
to the spin of the particle. As for the E(2)-like symmetry for a
massless particle, it is not difficult to associate the rotational
degree of freedom to the helicity. After some stormy history, it has
been now established that the two translational degrees of freedom
of E(2) correspond to the gauge degree of freedom.

Then the following question arises. Can the E(2)-like little group
for massless particles be obtained from the O(3)-like little group
by a group contraction procedure. The answer to this question is
YES. You may be interested in the following papers.

The story is not over yet. When we talk about gauge transformations on photons
we talk about one degree of freedom. There are two translation-like
degrees of freedom in Wigner's little group for photons. How can we collapse
those two into one. During the period 1985-91, I used to go to Princeton
regularly to tell Professor Wigner the stories he wanted to hear. In 1985,
he was 83 years old, but he was eager to write new papers. He was very
happy to hear that those two translation-like degrees correspond to
gauge transformation. He then raised the question of how two translations
become one gauge transformation.

We worked hard, and published a
paper in 1987 providing a solution to this problem. The point is
that the little-group is only iomorphic to the two-dimensional Euclidean
group. The little group takes the form of transformations on a
cylindrical surface consisting of rotations (helicity) and up-down
translations (gauge transformations).

Finally it is my pleasure to show my photos with the main characters of
group contractions. One is with Eugene Wigner and the other is with
Erdal Inonu.